Differential Equations and Their Applications Part 14 docx

250 Chapter 9. Numerical Methods of FBSDEs
Add the two inequalities above and apply Gronwall's lemma; we see that
sup(H~kI[ + IIckil) = V(h + At).
k
Applying the arguments similar to those in Theorem 2.3 we can derive the
following theorem.
Theorem 2.4.
Suppose that (A1)-(A3) hold.
Then,
-u(t,
z)] +
IV(~)(t,x) -u~(t,z)[}
= O(1).
sup { ]V(n) (t, x)
(t,~)
n
Moreover, for each fixed x E IR, U (n)
(.,
x) and V (~)
(., x) are
left-continuous;
for
fixed t C
[0, T],
U (~) (t, .) and V ('~)
(t, -)
are uniformly Lipschitz, with the
same Lipschitz constant that is independent of n.
w Numerical Approximation of the Forward SDE
Having derive the numerical solution of the PDE (1.5), we are now ready
to complete the final step: approximating the Forward SDE (1.4). Recall

that the FSDE to be approximated has the following form:
/0
(3.1)
Xt = x + s, Xs)ds +
~(s,
Xs)dWs,
where
b(t, x) = b(t, x, O(t,
x),
-a(t, x, O(t, x)O~ (t, x)) = bo
(t, x,
O(t, x), O~
(t, x));
~(t, x) = ~(t, x, o(t,
~)).
for (t, x) e [0, T] • IR.
To define the approximate SDEs, we need some notations. For each
n E IN, set At~ =
T/n, t ~,k = kAtn,
k = 0,1,2, ,n, and
n 1
(3.2) ~/n(t) =
Etn'kl[tn.~,t~.k+x)(t), t 9
[0, T);
k=0
gn(T) = T.
Next, for each n, let (U (n), V (n)) be the approximate solution to the PDE
(1.5), defined by (2.35) (in the special case we may consider only
u (n)
defined by (2.24)). Set

(3.3)
o~(t,x) = U(~)(T - t,x), on(t,~) = V(n)(T - t,x),
and
bn(t,x) =
bo(t,x, On(t,x),O'~(t,x));
~(t,x) =
a(t,x, On(t,x)).
By Theorem 2.4 we know that 0 = is right continuous in t and uniformly
Lipschitz in x, with the Lipschitz constant being independent of t and n;
w Numerical approximation of the FSDE 251
thus, so also are the functions ~n and yn. We henceforth assume that there
exists a constant K such that, for all t and n,
(3.4)
Ibn(t,x)
-bn(t,x')l +
I~n(t,x)
-~n(t,x')l _
KIx- x'l, x,x' e IR.
Also, from Theorem 3.4,
(3.5)
sup(b~(t,x)-b(t,x)l'+supl~n(t,x)-~(t,x)l
= O(1).
t,~ t,x
We now introduce two SDEs: the first one is a
discretized SDE
given
by
(3.6) 2/~=x+
g"(.,2?Lo(s)as + ~(.,2~),~(s)aw~,
where ~n is defined by (3.2). The other is an

intermediate approximate
SDE
given by
// //
(3.7)
X~ = x + b~(s, X2)ds +
Yn(s,
X2)dW~.
It is clear from the properties of ~n and ~n mentioned above that both
SDEs (3.6) and (3.7) above possess unique strong solutions.
We shall estimate the differences )(~ - X~ * and
X n - X,
separately.
LeInma 3.1. Assume
(A1) (A3). Then,
E{ sup IXT:-X:I 2} =0(1).
O<t<T
Proof.
To simplify notation, we shall suppress the sign "-" for the
coefficients in the sequel. We first rewrite (3.6) as follows:
/o //
2~ = No + u~ + bn(s, Xn)ds + an(s, f(n)dWs,
where
L /o
trbn,
f(n\
bn(s,22)]ds+
[an(.,2.~),
(~)_an(s,22)]dW~.
u~ =

L
k',
9
)n~(s)-
Applying Doob's inequality, Jensen's inequality, and using the Lipschitz
property of the coefficients (3.4) we have
E{ sup IX: - ~7:12 }
s<t
(~) _<~{ s~p i~nl ~ } + ~ ]/~{I~ n
-
~s~,~}~s
s<t
+ 12K 2
fE{lX$
-
J0
252
Chapter 9. Numerical Methods of FBSDEs
Now, set
as(t)
= E{ sups_< t [X~ - Xsnl2}. Then, from (3.8),
/o'
an(t) <
3E{ sup [un] 2 } + 3K2(T +4)
an(s)ds,
s<_t
and Gronwall's inequality leads to
(3.9) E~ sup[X n - Xsni 2 } <
3e3K~(T+4)E~
sup ]Uy]2~.

s<t ~ s<t J
We now estimate E{sups_< t ]uyl2}. Note that if s E
[tn'k,tn'k+l),
for
some 1 < k < n, then ??n(s) =
kAtn
(whence T - ~n(s) = (n - k)Atn, as
T = nat,)
and T - s E ((n - k - 1)Atn, (n - k)At~]. Thus, by definitions
(2.9) and (3.2), for every x E IR
0~(nn(~), ~) = ~(~)(T - nn(s), x) = ~(~)((~ - k)~t~, x)
: ur -
s,x) : O'~(s,x).
More generally, for all (s, x) E [0, T] x IR,
b'~(s,x) = b(s,x, On(s,x)) = b(s,x, On(~n(s),x)).
Using this fact, it is easily seen that
fo ~ b(v (~),X,o(~),O (v (~),
n -n n n -n
x,~(~))) - b(~,X n~ , O~(~,X:))
ds
/o'
~ { b(~n(s), f(vn~(~), On(s,
2~n~ (~))) -
b(s, X2, 9~(s,
2v~(~)) )
b ~ n -n __
+ (~,x~,o (~,x,o(~))) b(~,x:,on(~,x:)) }d~
=11 + I2.
Using the boundedness of the functions bt, b~ and
by,

we see that
{
~
~/o' { lt~ll~l~~ § H~JJ~I~,~-~:1}~,
/o'
I2 < K]lbyH~ . If(vS(s) - X2]ds.
Thus,
- + x jlds,
where h" depends only on
K, ]]btllc~, IIb~]l~
and Ilbvll~. Since
inn(s) -
sJds = (s - tk)ds < -~ k=o
w Numerical approximation of the FSDE 253
E{sup~<~ fo~b ~'~, 2 ~. ),~(s)- b~(8, X:)ds 2 }
(3.10)
0 t
T4
Using the same reasoning for a with Doob's inequality, we can see that
~{sup [ ~n(. X.~)~(~) _
~~ X:)eW~ 2}
u<t
(3.11) _< 8/~2 {
EIy;n(~)-XFI2ds+ (s-~n(s))2ds}
< 8~2{ fo ~
Combining (3.10) and (3.11), we get
fo t 16)1
E{sup
I,,~12} < ~(4T + 16) E[2~(~) -
X~[2ds + R~T(T + 3 ~2"

s<t
Thus, by (3.9),
.{ su8 ,~: - x:l ~ }
(3.12)
<_ 3e 3K2(T+4){/~2(4T
+ 16)
EI2,~.(~)
- X2[2ds
+K2T(T+~)n~ }-
Finally, noting that ] ~-(s) - X~I < I ~(~) - -~21 + 122 - X21 and that
we see as before that
fo EI2~(~)
-
2212ds <_ 2
IlbllL(s - v~(s)) 2 + [l~llLIs - v~(s)l
ds
< 2]lbll~T 1 1
- 3 n 2 +II~II~T n
Therefore, (3.12) becomes
i
t -n n 2
(3.13) E{suplXn x:I 2} <C, +C2z~+C3
f E#suPlX~-X~l
}as,
s<t It Jo t. r<s
where C1, C2 and Cz are constants depending only on the coefficients
b,
a and K and can be calculated explicitly from (3.12). Now, we conclude
from (3.13) and Gronwall's inequality that
~n(t) <_ /3ne CT,

Vt 6 [0, T],
254 Chapter 9. Numerical Methods of FBSDEs
where
/9~ = Cln -1 + C~n -2
and
CT = C3T.
In particular, by slightly
changing the constants, we have
an(T):E~ sup
IX:-Xnl 2}
< C, + 02 =0(i),
- 0<~<~ -~-
proving the lemma. []
The main result of this chapter is the following theorem.
Theorem 3.2.
Suppose that the standing assumptions (A1) (A3) hold.
Then, the adapted solution
(X,
Y, Z) to the FBSDE (1.1) can be approxi-
mated by a sequence of adapted processes
(X "n, Y~, Zn),
where f(~ is the
solution to the discretized SDE (3.6) and, for t 6
[0, T],
~n :: 8~(t,2tn); Z? :=
-a(t,2~,sn(t,f~))O~(t,f(?),
with O n and 0 n being defined by (3.3) and U (n) and V (~) by (2.34). Fur-
thermore,
(3.14) E{ 0<t<TSUp
]f(: XtI+O<t<TSUp

]~n Ytl+0<t<Tsup
I'~-Ztl}=O(~n).
Moreover, if f is C 2 and uniformly Lipschitz, then
for n large
enough,
(3.15)
E{f(2~,
2~)} -
EU(XT,
Z~)}[ _<
K
n
for a constant K.
Proo].
Recall that at the beginning of the proof of Lemma 3.1, we have
suppressed the sign "-" for b and ~ to simplify notation. Set
~n(t) = { sup Ibm(t, x) - b(t, x)l 2 + sup lan(t, x) a(t, ~)l ~
},
x x
where b, b n, a and a n are defined by (3.1) and (3.3). Then, from (3.5) we
know that sup t
Izn(t)l
= O(~A~). Now, applying Lemma 3.1, we have
~{
:~ I~: - ~J~}
_< ~{
~u~ i~:- ~:l ~ }. ~{ ~u~ i~: - ~sl ~ }
w Numerical approximation of the FSDE 255
Further, observe that
<_4T fot Elbn(s, X2) - bn(s, X~)[2ds

+ 16
Elan(s, X2) - a n
(s,
Xs)12ds +
4(T + 4)
r
~4(T + 4)K ~ E{ sup IX~ - X~I ~}es + 4(T + 4) ~n(s)e~.
r<_s
Applying Gronwall's inequality, we get
{ } Jo
(3.16) E sup
[X• - Xs[ 2 <
4(T + 4)
Sn(s)ds"
e 4(T+4)K2 <
n- ~,
s<t
where C is a constant depending only on K and T. Now, note that the
functions 0 and On are both uniformly Lipschitz in x. So, if we denote their
Lipschitz constants by the same L, then
0<t<T
_
on(t
2n~121
<
2E~
sup
lO(t, Xt)- ,
, t
:~ I

0<t<T
+ 2E{ sup 10~(t,22)-0(t,~)l 2}
0<t<T
0<t<T
(t,x)
by Theorem 3.4 and (3.16). The estimate (3.14) then follows from an
easy application of Cauchy-Schwartz inequality. To prove (3.15), note that
Theorem 2.3 implies that, for n large enough, snp(t,x)10n(t,x)
-0(t,
x)l
Cn -1,
for some (generic) constant C > 0. We modify )(~ as defined by
(3.6) by fixing n and approximating the solution X ~ of (3.7) by a standard
Euler scheme indexed by k:
f0 f0
2~ 'k = x + b(.,2.~,k)n,c(s)ds + a(.,2?'k),,,(s)dWs.
It is then standard (see, for example, Kloeden-Platen [1, p.460]) that
(3.17)
C1
E{:(X~)} - E{f(2~'k)} <_ K
256 Chapter 9. Numerical Methods of FBSDEs
On the other hand, we have
Ig{f(XT)}
- E{f(X~)}[ <_ KE{IXT -
X~[ }
(3.18)
C2
O~t~T ) ?2
for Lipschitzian f, by (3.16). Therefore, noting that X~ as defined by (3.6)
is just _~n,n

t , the triangle inequality, (3.17) and (3.18) lead to (3.15). []
Comments and Remarks
The main body of this book is built on the works of the authors, with
various collaboration with other researchers, on this subject since 1993.
Some significant results of other researchers are also included to enhance
the book. However, due to the limitation of our information, we inevitably
might have overlooked some new development in this field while writing
this book, for which we deeply regret.
In Chapter 1, the results on the pure BSDEs, especially the fundamen-
tal well-posedness result, are based on the method introduced in the seminal
paper of Pardoux-Peng [1]. The results on nonsolvability of FBSDEs are
inspired by the example of Antonelli [1]. The well-posedness results of FB-
SDEs over small duration is also based in the spirit of the work of Antonelli
[1]. The whole Chapter 2 is based on the paper of Yong [4].
In Chapter 3 we begin to consider a general form of the FBSDE (1)
with an arbitrarily given T > 0. The main references for this chapter
are based on the works of Ma-Yong [1], virtually the first result regarding
solvability of FBSDE in this generality; and Ma-Yong [4], in which the
notion of
approximate solvability
is introduced. A direct consequence of the
method of optimal control is the Four Step Scheme presented in Chapter 4.
The finite horizon case is initiated by Ma-Protter-Yong [1]; and the infinite
horizon case is the theoretical part of the work on "Black's Consol Rate
Conjecture" presented later in Chapter 8, by Duffie-Ma-Yong [1].
Chapter 5 can be viewed either as a tool needed to extend the Four Step
Scheme to the situation when the coefficients are allowed to be random, or
as an independent subject in stochastic partial differential equations. The
main results come from the papers of Ma-Yong [2] and [3]; and the appli-
cations in finance (e.g, the stochastic Black-Scholes formula) are collected

in Chapter 8.
The method of continuation of Chapter 6 is based on the paper of Hu-
Peng [2], and its generalization by Yong [1]. The method adopted a widely
used idea in the theory of partial differential equations. Compared to the
Four Step Scheme, this method allows the randomness of the coefficients
and the degeneracy of the forward diffusion, but requires some analysis
which readers might find difficult in a different way.
Chapter 7 is based on the work of Cvitanic-Ma [2]. The idea for the
forward SDER using the solution mapping of Skorohod problem is due
to Anderson-Orey [1], while the Lipschitz property of such solution map-
ping is adopted from Dupuis-Ishii [1]. The proof of the backward SDER
is a modification of the arguments of Pardoux-Rascanu [1], [2], as well as
some arguments from BuckdahmHu [1]. The proof of the existence and
uniqueness of FBSDER adopted the idea of Pardoux-Tang [1], a general-
ized method of contraction mapping theorem, which can be viewed as an
independent method for solving FBSDE as well.
258 Comments and Remarks
Chapter 8 collects some successful applications of the FBSDEs devel-
oped so far. The integral representation theorem is due to Ma-Protter-Yong
[1]; the Nonlinear Feynman-Kac formula is in the spirit of Peng [4], but the
argument of the proof follows more closely those of Cvitanic-Ma [2]. The
Black's consol rate conjecture is due to Duffie-Ma-Yong [1]; while hedging
contingent claims for large investors comes from Cvitanic-Ma [1] for uncon-
straint case, and from Buckdahn-Hu [1] for constraint case. The section on
stochastic Black-Scholes formula is based on the results of Ma-Yong [2] and
[3], and the American game option is from Cvitanic-Ma [2].
Finally, the numerical method presented in Chapter 9 is essentially
the paper of Douglas-Ma-Protter [1], with slight modifications. We should
point out that, to our best knowledge, the scheme presented here is the
only numerical method for (strongly coupled) FBSDEs discovered so far,

and even when reduced to the pure BSDE case, it is still one of the very
few existing numerical methods that can be found in the literature.
In summary, FBSDE is a new type of Stochastic differential equations
that has its own mathematical flavor and many applications. Like a usual
two-point boundary value problem, there is no generic theory for its solv-
ability, and many interesting insights of the equations has yet to be dis-
covered. In the meantime, although the theory exists only for such a short
period of time (recall that the first paper on FBSDE was published in
19930, many topics in theoretical and applied mathematics have already
been found closely related to it, and its applicability is quite impressive.
It is our hope that by presenting a lecture notes in the series of LNM,
more attention would be drawn from the mathematics community, and the
beauty of the problem would be further exposed.
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Index
A
adapted solution, 10,25,138
approximate, 54
classical, 105
strong, 105
weak, 105
adjoint equation, 4
aggregator, 5
allocation, 6
a-efficient, 6
American game option, 232
appreciation rate, 7,208
approximate solvability, 54
B
Bellman's principle of optimality, 58
Black-Scholes formula
stochastic, 228
robustness of, 231
Black's conjecture, 202
bond, 7

bridge, 139
BSDE, 3
BSDER, 171
BSPDE, 103
Brownian motion, 1
C
charasteristics, 237,247
comparison theorem, 22,130,214,217
condition
monotone, 151
Novikov, 23
consumption, 7,208
contingent claim, 7
control, 3,33,52
controllability, 33,52
cost functional, 3,53
current utility function, 5
D
dynamic programming method, 57
differential utility, 5
E
equation
adjoint, 4
Hamilton-Jacobi-Bellman, 60
Riccati, 46
state, 3
stochastic differential, 1
backward, 3
forward-backward, 1,4
European option, 7

exercise price, 7
expiration date, 7
F
FBSDE, 1,4
FBSDER, 181
formula
stochastic Black-Scholes, 228
Feynman-Kac, 197
FSDER, 169
Four Step Scheme, 81
function
representing, 90
utility, 5
value, 58
G
Gelfand triple, 111
H
Hamiltonian, 59
hedging strategy, 8,209
I
indicator, 175
L
large investor, 207
M
maturity date, 7
method of
continuation, 137
optimal control, 51